Le Nozze Di Figaro: What Do They Want?

Uncategorized | Posted by Brian PCF
Jan 26 2014

The Count and Countess in Le Nozze di Figaro present characters with two very opposing and dramatically different personalities.  The Count is distinguished but a buffoon and an adulterer.  The Countess is refined and cunning while at the same time hopelessly romantic. Mozart brings these characteristics to the surface right out of the gate, but refines and adds depth as the opera progresses.  The core questions to examine with respect to these characters is: what do they want?

Part I:  The Count, Figaro and Susanna

 

The Count doesn’t take the stage until No. 6 but before he shows up we already know a great deal about him.  Susanna informs Figaro in No. 2 that the reason that she and Figaro have been provided with lavish new quarters as a gift for their wedding is because the Count wants Susanna close enough for him to slip in and carry out an affair.  It’s worthwhile to examine Susanna and Figaro closely as they bring out greater depth in the Countess and the Count.  The conflicts that Figaro and Susanna go through are a bright counterpoint to the conflicts of the Count and Countess.

Very unlike the interaction of the Count and Countess later in the play, after Susanna has informed Figaro of the Count’s plans in No. 2 and Figaro doesn’t seem to understand her, Susanna mocks him and Figaro gently deflects it:

 

S: Because I am Susanna, and you are insane.

F: Thank you, spare the compliments.

 

In No. 1 we already see what sounds like a “normal” husband/wife conflict that comes to a quick resolution between Figaro and Susanna. He wants to measure the place for their bed, while she wants him to pay attention to her and talk of their wedding.  This is resolved quickly and within a few measures Figaro relents and the couple sing of the happiness they feel towards their upcoming marriage.  This piece is an uplifting duettino sung in G-Major.  We see the same key in No. 8, No. 12, No. 14 and No. 21.  All these parts have a common theme of playfulness and affection.  This is how these two will act and react throughout the opera.  Susanna has a temper (which sometimes erupts in a slap to Figaro’s face) but every time this happens Figaro knows the right words to sooth his beloved and they quickly make up.

Some of the conflicts that face the Count and Countess have already been laid out before either show up.  The Count is trying to carry out an affair with his wife’s maid servant and Figaro having found this out has pledged vengeance on the count through what seems a strange set of tactics: “the arcane art of concealing and taunting.”  Further, the Count is in a difficult position politically.  He has done away with droit du seigneur in name, but still wants to carry it out in fact.  Figaro makes great use of this by enlisting the countrymen and women to celebrate this “noble deed” in No. 8.  The Count also has unreliable subordinates in the person of Basilio, who not only fails to carry out his wishes to woo Susanna for the Count, but also has a tendency to talk too much.

Lastly, there is Cherubino the page.  He will be a thorn in the side of the Count throughout the opera.  One of the most intriguing aspects of the relationship between the Count and Cherubino is that the page is a constant problem for the Count without really trying.  While Figaro and Susanna (and later the Countess and Marcellina) all work against the Count actively, Cherubino seems to cause trouble for the Count often just by being himself.

Before the Count ever arrives on stage, Cherubino has already been banished from the castle for haunting the quarters of one of the Count’s servants and sometimes bedfellow, Barbarina.  Rather than flee, Cherubino tries to enlist Susanna in delivering a canzonetta to the Countess and to ask her to intervene on his behalf.

When the Count does enter in No. 6, we see what he wants very clearly: Susanna.  His plan is straightforward and while Basilio has laid the groundwork the Count makes his suit very clear: accept the right of droit du seigneur and the Count will pay her.  It’s implied if she accepts the trip to London with the Count he will continue to be her benefactor.

But the Count’s plans are foiled by Cherubino again.  Basilio arrives and having heard that Cherubino was headed to her quarters tells Susanna to make the page tame his obvious affection towards the Countess because everyone in the castle has noticed.  Basilio doesn’t know the Count can hear him, and when the Count reveals himself Basilio is strangely apologetic and backtracks on his statement.  In this very same scene it’s made obvious that Basilio delights in palace intrigue.  So his seeming (both through the libretto and Count’s low and forceful vocals a counterpoint to Basilio’s high and deferential vocals) fear of what the Count will do with this new knowledge stands out against his normal behavior.  Now that Cherubino has inserted himself unwittingly in this situation, he has caused the Count to think that someone is trying to seduce his Countess.  The Count then changes his main goal and becomes determined to keep the Countess true to him.

An item that is interesting to mention at this point is the way that Mozart and De Ponte admirably shape the plot and music in this first part.  While we’ve focused on the conflicts that face the Count, the way they are presented thus far it’s really unclear who will come out on top.  While we assume in “Le Nozze di Figaro” that our protagonist will get a happy ending, there’s enough dramatic tension deftly generated that one really must wait for the denouement to figure out the winners and losers.  And even then, the future is unclear.

This leads us to attempt a first blush summary answer to our initial question as least as it relates to the Count: what does he want and why?  It seems thus far that he wants to be an unchallenged bull in the stableyard.  He banishes Cherubino at first for sharing Barbarina and again for his attention to the Countess.  He is planning to take his officially relinquished right of droit du seigneur with Susanna (which necessitates Figaro being absent) and plans to continue that relationship regularly in London, where he has promised to take Susanna and ensure Figaro is busy enough to leave them time alone.  The Count wants lone sexual dominance within his castle.  However, while Cherubino, Figaro and Susanna have started to take action against the Count’s designs they are not as effective as the Countess’ plans or her force of will.

Part II: The Countess

 

We meet the Countess in the opening scene of Act II in a simple aria filled with sadness and longing.  We are in E-flat here, the same key that Cherubino sang to Susanna in No. 6.  What is important to note here is that in Cherubino’s aria the music is longing, yet hopeful.  This fits with a character who has seemingly had great success in love.  In the Countess’ aria there is definitely longing but not a great deal of hope.  The whole part is larghetto, or “a fairly slow tempo”, the first slow tempo we’ve seen in the opera.  Everything else has been played with speed for either comedic effect or reflecting the bravado of the characters.

What’s interesting in comparing the libretto in No. 6 and No. 10 is Cherubino’s description of his feelings as “I know longer know what I am, what I do; now I’m all fire, now all ice.”  The Countess’ refrain throughout No. 10 is a similar contrast: “give me back my treasure, or at least let me die.”  The choice of phrasing and of making both parts in E-flat seem to signify there is a similarity but a seemingly great difference comes through musically between a youthful love and a mature love.  A love with many opportunities in front of it, and a love that should it fail will leave the Countess with not only a terrible grief but a continued torturing of her heart as she must live as the Count’s bride but suffer him flaunting his dalliances.

The instrumentation in No. 10 is the simplest we’ve seen thus far, but also very meaningful.  It seems that Mozart and De Ponte want to focus mainly on the libretto and vocals to communicate the feeling of this piece.  One can’t help but empathize with the Countess here. We’ve seen how dastardly the Count can be in Act I, and now we hear how the Countess’ reacts and how she suffers because of the Count.

There is one interesting piece of instrumentation in the clarinet and bassoon that may give us some hope for the Count and Countess though.  Between each verse, the clarinet and and bassoon play a short duetto.  It begins in the 27th measure of No. 10 with just the clarinet playing a short tune:

 

Then right after in the 29-31st measure, the clarinet comes in again, but the bassoon picks up on the melody and mimics it:

We see this once more in the last three measures of the part where we find the clarinet starting and the bassoon answering again, and then completing the piece with a lovely finish: both playing a G and an A because the clarinet is in B-flat on the 2nd to last notes and then finishing with a lovely combination of clarinet and bassoon in harmony.

If one could look at these short duetto’s within the piece, and specifically notice that they are played while the vocalist is silent, how could one not be a bit hopeful for the Countess’ future?  It seems plausible that these pieces are put in to counterbalance the slow longing piece with a bit of hope that if the Countess takes the lead, the Count will come around.  It seems from this piece that what the Countess wants is clear: to regain the love of her husband.

 

Part III: Singing Together

 

We’ve now seen how the Count and Countess sing apart, but what of them singing together?  Our first opportunity for this is in No. 12 and No. 13.  So far the Count has, either out of a sense of possessiveness or fear of being cuckolded, launched into action to determine if the Countess is being untrue.  At the same time, Figaro launches a plot to further the Count’s suspicion of his Countess’ potential infidelity by putting a scandalous note where Basilio can find it and inform the Count.

The Count returns from a hunt (supposedly, interesting timing…) with the note, and he chooses a bad time to do it as the Countess has Cherubino half naked and alone in her quarters as part of her plot to ensnare the Count.  There is certainly a hint in the libretto here that there is potential for the Countess to reciprocate Cherubino’s affection for her which would only inflame the Count more and adds to the dramatic tension.  While the Countess shuttles Cherubino off into her closet and tries to persuade the Count she is alone, Cherubino (unwittingly again) bumbles into something in the closet and makes a noise.

The Count’s possessiveness is on display here but also his concern over propriety; his wife hits upon it right away.  When the Count demands that she open the closet door, the Countess replies “A scandal, an uproar can still be avoided, I beg you.”  This buys the Countess some time as the Count doesn’t want to involve the servants and so must hunt for the necessary tools to open the door himself.

When the Count returns and the Countess admits that it is not Susanna within, the Count turns deadly.  He demands from the Countess: “What is it then? Say…or I’ll kill you!”  When she admits it’s Cherubino his rage has totally taken hold.  We see here why Figaro wanted to confuse the Count with tricks: when the Count feels played upon he goes into a fit of rage that bedevils him.  We see this in a bit of comedy, when the Countess tries to defend Cherubino:

 

Countess: He is innocent, you know it…

Count: I know nothing!

 

The Count subjects have succeeded in angering and confusing him.  Death and vengeance are the only thing on his clouded mind.  The libretto is I believe purposely vague in the last few measures.  The Count simply repeats “Mora, mora” which could mean that he will kill the page or it could mean he will kill the Countess as well.  Probably the Count doesn’t even know what he will do.

When Susanna comes out of the closet, the Count becomes docile quickly.  One wonders why the sudden change.  Is he embarrassed because he thinks he’s been tricked and goaded into this rage by the Countess and Susanna?   Has his rage subsided so quickly because not only is Cherubino nowhere in sight, but also because of his sense of decorum?  Can he not let himself lose his temper in front of a servant girl, even one he is trying to bed?  If true, how much of this drives his begging the Countess’ pardon and his profession of love to her immediately following this?

Important to what  follows in the very next scene between the Count and Susanna is the idea first introduced at this point in the opera by the Countess that the Count is a “cruel man”, or “crudele”.  The Countess repeats it several times, and the Count continues to beg her forgiveness, even stooping so low to ask for Susanna’s assistance.

Part IV: The Countess takes the Offensive

 

The first three things to point out in the pivotal scene in No. 16 is first, the possibility of the Count becoming self aware.  Though a short line in the recitative, after looking back on what has happened he says: “and for my honor…honor…where in the devil has human error carried me!”  So many possible meanings here: is he thinking about his adultery?  Or is he castigating himself for not being able to figure out how he is being manipulated?

The second thing to notice is the Countess continues to press her plot to catch the Count in flagrante delicto but changes her bait from Cherubino to Susanna.  The question to ask here is why and what does she want?  It seems as if she has thrown off most of the suspicion surrounding her (she could surely overcome Antonio’s not-so-lucid eye witness account) and the Count has professed his love and asked her pardon.  Why press on?

The third thing to notice is how the Count’s choice of words echo the Countess’ in the previous scene.  In it, the Count’s refrain is “perche crudel?”, “why are you so cruel?”  But this time it is he speaking of Susanna’s denying his repeated advances.  It almost seems that the Count and Countess have become so accustomed to each other, they share the same taste in language.

What a fascinating relationship we have so far between the Count and Countess.  She knows his hot spots: the Count’s fear of embarrassment, of being cuckolded, and his adulterous nature.  Yet she hasn’t tamed the bull yet.  As soon as Susanna lays the Countess’ trap for the Count in No. 16 claiming she’ll agree to his offer of money for her favor, he jumps at the chance.

This is one of the few parts that is played in a minor chord.  We start in the A-Minor before transitioning to the C-Major and finally landing in A-Major.  The only other times minor keys are used are briefly in No. 22 and then most notably in Barbarina’s aria in No. 23.  The Count sings in the minor key to start No. 16 and sings of how cruel Susanna has been to him and demands to know why she has made him wait so long.  Barbarina sings of a lost pin, which most likely represents her lost innocence, and this is a longing much more powerful than the Count’s in No. 16.  Perhaps we can gain a greater appreciation for these characters by comparing these two pieces to the Count’s aria in No. 17 and the Countess’ aria in No. 10, respectively.

In No. 23, Barbarina’s aria, there is profound sadness and yearning very similar to the Countess’ aria in No. 10.  But it is played in the minor key, just like the beginning of No. 16 when the Count longs to know why Susanna has denied him for so long.  It seems that these two pieces, No. 16 and No. 23 go together somehow.

And when the Count gets his aria in No. 17, there is nothing about love.  Here he only wants vengeance against those who have wronged him.  This part is in D-major, the main key of the play so it seems that we are to pay close attention to this point.  This is also the only aria the Count sings.  If we compare this to the Countess’ aria, we see what a hopeless romantic she is compared to the Count.  Both in No. 10 (and her other aria No. 19), she says she cannot control her love.  If she can’t have his love, she wants to die and wishes she could forget the love they shared at one time.

This brings us to an interesting point about the play as a whole: Do any of the main male characters deserve our empathy?  Are any of them capable of love?  Basilio in his aria in No. 25 comes right out against it and says to wear a donkey’s hide to protect oneself from it.  In No. 18, Bartolo agrees to marry Marcellina, but only because they have found their son is Figaro.  The Count’s trespasses against his wife’s love have taken up most of the opera thus far.  What of Figaro?  He seems happy at the idea of marrying Susanna from the very beginning.  However, he is a comedic figure and he continues to distrust Susanna until the very end of the play.  So we are left with only Cherubino, whose whole person burns with love.  Is he even a real person or as Basilio describes him in No 7, is he truly “loves’ cherubin”?

On the other hand can the women in the play do anything but offer forgiveness and love?  Simple Barbarina wants to marry Cherubino, the castle Don Juan, yet still accepts the Count’s favors as well.  Marcellina wants to buy a marriage to Figaro, a man much younger and of seemingly ignoble birth.  Susanna must suffer through Figaro and the Countess’ many plots and schemes (and on her wedding day!) to finally succeed in marrying her man.

And then we have the Countess.  What immense love she must bear the Count to put up with his behavior.  He carries out his dalliances in almost plain sight.  He must notice his wife’s suffering and yet he ignores it.  She will stoop so low as to catch the Count in the act of being unfaithful if that’s what it takes to recapture (or so she hopes) his love.

These points are here mainly to bring greater relief to the question: What do the Count and Countess want and why?  In No. 28, we see the culmination of all of the characters who schemed and all who loved.  Now perhaps in the end the Count becomes infected with love courtesy of Cherubino.  He is pricked in the finger by a pin with Cherubino’s blood on it and he is kissed by Cherubino in No. 28 when the page is trying to steal a kiss from the Countess.

We find ourselves reviewing the whole of the opera here during No. 28 musically with five key changes.  All in major chords, we go from: D-G-Ef-Bf-G-D.  Each of these evokes certain parts that have come before to echo the same feelings of the characters in their parts prior to the opera’s finale.

The key moment to discover what motivates the Count and Countess though is when the Countess reveals herself at the very end.  Susanna (in the Countess’ dress) is accused of perfidy with Figaro and begs the Count’s pardon in front of his subjects and the Count denies them vehemently: “no, no, no, no, no, no!”  When the Countess reveals herself and the Count realizes he’s been duped it seems that finally we know what the Count and Countess want: he wants her forgiveness for his behavior, and she wants to be asked for it.

 

Comparing Axiomatic Systems: Euclid & Lobachevsky

Uncategorized | Posted by Brian PCF
Nov 26 2013

Geometry seems a unique science. Those who practice what we would call “Natural Sciences” gather independent facts, try to find patterns, and then through cumbersome intermediate steps determine laws. On the other hand as geometry progresses in complexity everything that it creates is a new law, equally important from first to last. So long as it does not contradict anything that’s come before it is considered true no matter what its basis. Thus proving any system of geometry turns into simply avoiding contradicting oneself. Euclid did this elegantly by asking us to grant him the “Parallel Postulate” and then not using it for twenty eight propositions. Lobachevsky asks us to do this by proposing that the angle of parallelism is less than the sum of two right angles. Both of these systems seems true in that they do not contradict themselves, but they certainly contradict one another.

How does comparing these two systems inform us as to the necessity of questioning what we consider “laws” and what is the relative value of studying laws for their own sake versus questioning their “trueness” in comparison with other systems of laws? Said another way, what is the value of using our deductive reason to analyze deductive systems and our inductive reason to compare deductive systems?

Part I: Using deductive reason to analyze deductive systems.

Let us examine which deductions are shared by both Euclidean and Non-Euclidean systems and why. It seems implied that any of Euclid’s work still holds in Lobachevsky except for the fifth postulate (which gets us to the angle of parallelism being equal to two right angles) and anything that Lobachevsky specifically redefines. It must be different in some basic concepts otherwise there are no points in Lobachevsky. Where Euclid starts with a point, Lobachevsky starts immediately with the line and since he uses a different definition, this must be different than a Euclidean line:

Lobachevsky Theorem 1: A straight line fits upon itself in all positions. By this I mean that during the revolution of the surface containing it the straight line does not change its place if it goes through two unmoving points in the surface: (i.e., if we turn the surface containing it about two points of the line the line does not move.)

For Euclid’s definition of a line to work, we must include his definition of a point as well:

Euclid Definition 1: A point is that which has no part.
Euclid Definition 2: A line is breadthless length.

So it is now spelled out in the very beginning of Lobachevsky that rather than the Euclidean universe that is made up exclusively of points and lines and the relationships between the two, we start with a line and a surface with the potential for rotation as the fundamental building blocks of Lobachevsky’s geometry.

As Lobachevsky builds his system, we see several different concepts in his definitions, but none are that far afield of what is true within Euclid’s system until we get to Theorem 16 and find out what he calls the angle of parallelism can be either what Euclid states it is: the sum of two right angles, or possibly less than that. What he mathematically represents as:

If Euclid: ∏(p) = ∏/2.
If not: ∏(p) < ∏/2.

So at this point, we take nothing for granted within Lobachevsky’s deductive system but the difference between him and Euclid is now becoming clear. In terms of pure deduction, he could just as easily propose that ∏(p) > ∏/2 or some random series of integers like ∏(p) = 4.2114591913951. At this point the key to hurdling the questions within pure deduction will be his ability for all resulting theorems to be consistent with his “if” statement.

As we continue to build to his true break from Euclidean geometry in Theorem 22, we find that sum of the angles of a triangle will be the same as his angle of parallelism in Theorem 16. That is, that the sum is less than ∏/2. Again he gives two options here, the Euclidean and what he calls the “imaginary geometry” which he says can apply to both rectilineal and spherical triangles.

However, the big deductive question in this is evidenced by Theorem 24 where he proposes that “The farther parallel lines are prolonged on the side of their parallelism, the more they approach one another.” Since we already have Theorem 17 “A straight line maintains the characteristic of parallelism at all its point” and Theorem 2 “Two straight lines cannot intersect in two points” we must throw out the most obvious question in Lobachevsky: do his straight lines curve? Then we must logically ask if all of his surfaces curve in some way? However, it seems in Theorem 22 that since he specifically differentiates between the rectilineal and the spherical triangles, his surfaces are consistent with the surfaces found in Euclid.

So we are only left with a few options in our deductive reasoning: first, that Lobachevsky’s prose can accurately describe a model that diagrams cannot. Second, we can wonder if this is similar to the parable of Meno’s ass: that if one line continues to move closer to the other line by halves, they will never meet.

Here we are at an impasse in our deductive capabilities. Both systems seem true in and of themselves, but both come to different conclusions about fundamental parts of geometry. It seems that to understand how true these systems are we must compare them in more detail to one another and use our inductive reasoning as much as possible.

Part II: Using inductive reason to compare deductive systems.

Geometry is pure deduction: everything follows from what comes before. To say this in another way is to restate our opening proposition: geometry seems very different than the natural sciences. Bacon called upon the natural sciences to use its inductive reason to find true laws, and while human nature has the same propensity towards error as in Bacon’s time, science has progressed and found rapidly appreciating application in understanding and making use of nature’s laws.

So how do we go about using our inductive reason to analyze these two systems of geometry? The first thing we can do is ask ourselves how to go about it: just because something that is designed to be purely deductive shows no obvious deductive flaws, how can we use to use inductive axioms to compare these deductive systems. Let us first look at an inductive axiom that seems to meet Bacon’s tests and has held up to several inductive and deductive challenges since its first introduction: Darwin’s survival of the fittest by means of natural selection.

How does Lobachevsky hold up against Euclid (and vice versa) in the application of this axiom? Though it is outside the scope of this paper, it would seem that we are uniquely fortunate in trying this model of examination because rather than large historical gaps being present through missing fossils, we see every deductive step in Euclid and Lobachevsky’s work numbered out for us. We can look at those individual evolutions of the species and then look at how it affected the larger genera of science. We can rather simply ask the question (the answer may not be so simple), do these systems act within science as a dominant species or one that looks like it will become extinct? As Darwin describes the difference: “new and improved varieties will supplant and exterminate the older less improved and intermediate varieties; and thus species are rendered to a large extent defined and distinct objects. Dominant species belonging to larger groups tend to give birth to new and dominant forms.”

While Euclidean geometry helped give birth to Newton’s physics, it was the Non-Euclideans (including Lobachevsky) that gave birth to Reiman which helped produce Einstein’s Theory of General Relativity which showed the flaws in Newton’s formulation of gravity and the movement of celestial bodies.

What is interesting and brings us full circle in a sense is that both are still accepted as “true” within certain parameters. When describing a body moving through space, so long as that body is moving in a unified gravitational field and nowhere close to the speed of light, Newton’s system works flawlessly. If we need to predict the movement of a body through space with inconsistent gravitational field and/or moving close to the speed of light however, we must use Einstein’s model to make an accurate prediction.
So it seems that even in the natural sciences, there are models that are sometimes true and sometimes less true.

Conclusion:

What one can potentially learn from comparing the two are first: that even that which stands for centuries and is taught as doctrine must be tested, changed, and potentially improved upon. If we do not attempt this we not only limit ourselves to profound discoveries that may aid centuries of scientific development, we also contradict what Bacon warns us against as the greatest pitfall to human understanding: “Those who have taken upon them to lay down the law of nature as a thing already searched out and understood, whether they have spoken in simple assurance or professional affectation, have therein done philosophy and the sciences great injury.”

Second is proof of the concept that there can be more than one answer. Euclidean geometry still “works” in a Euclidean model. Non-Euclidean Geometry works in a Non-Euclidean model. Both are tools that have applications in different fields and have value in and of themselves.
____________________________

Reference:
Full text of Euclid’s Elements here.
Full text of Lobachevsky’s Theory of Parallels here.

The Structure of Euclid’s Elements, Book I

Uncategorized | Posted by Brian PCF
Nov 11 2013

The structure of Euclid’s Elements can be explained simply by “lack of chaos”. Everything that comes about does so orderly and methodically. There is no new thing. There is no room for the addition of a new element, like in the periodic table. There is no new word that can be invented like Shakespeare did for language. Everything is just either a point, a line, or something brought about by the relationship of the two. There is no discovery of something new in that sense in Euclid, only something new in terms of the relationships between points and lines.

This leads to the second great structure or theme of Euclid’s Elements: his invitation to find flaws or explain it better. The arguments are so elegant, they are so “obvious” once they are explained, an amateurish mind might think “I can prove that wrong” or “I can do that better”. Euclid leaves that avenue open by explaining every point succinctly and referencing every conclusion exactly.

Section I: The Definitions

The definitions provide a type of alphabet for Euclid’s language. All of the propositions that he constructs will share these basic building blocks. What’s most striking about the first two definitions is that they are defined by what they are not:

1. A point is that which has no part.
2. A line is breadthless length.

These first two definitions demonstrate that Euclid’s universe is very much abstract: meaning it can’t exist in a real physical world that we can see or construct, only in our imagination. And yet the construction of the remainder of the definitions, and hence all of the propositions as well, rely on these two things: lines and points.

So two question to return to are: a) is Euclid creating an entire system that depends on us believing that things exist that we cannot prove exist and/or cannot physically comprehend, and b) do we accept a system that must be tied together using things that can’t exist in the physical universe as a starting point? While these questions sound similar, we will see they lead us to different conclusions.

But what are the definitions themselves? They are a type of relational shorthand. They are all made up of points and lines but in order to simplify his terms Euclid creates (or at least clearly defines) concrete descriptions based on relations between points and lines. These definitions serve a similar role to early propositions: once we have an agreed upon conception of these ideas, we can simply refer to them and we will understand their meaning specifically and in the context of their use.

Section II: The Postulates

The postulates, or one translation: “grant me this”, help move the definitions from abstract to real or at least easier to understand. It is interesting that Euclid uses different verbs for each of the first three postulates:

1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center or distance.

Why not just use “draw” or “produce” or “describe” for each postulate? Why use different terms? The most likely answer is because Euclid thinks to make these different things more understandable, we must do specific things to make them so. Producing or drawing a circle will not do, it must be “described.” Drawing or describing a straight line continuously in a straight line cannot be done, it must be “produced”. Likewise with a straight line from any point to any point, it must be “drawn”.

While this logic seems circuitous, one finds the difference necessary once within the propositions. Without them, their use and definitions become less clear. The most obvious of example is the circle. To “draw” or “produce” a perfect circle is something commonly regarded by artist’s as impossible. But we can create something, say “this is a circle” and then with our definitions mesh this with the context and meaning within the proposition.

Section III: The Common Notions

The Common Notions might be easier understood if Euclid called them “Common Sense Notions”. These are things which if not indisputable, would be very hard to dispute. One can imagine that Euclid created the whole of the Elements, but some competing school of thought decried his work for not laying out, for instance, that a “whole is greater than the part”. So while these things are obvious, the theme of construction tightly woven with what’s come before must be satisfied and hence these common sense sayings or notions must be included.

Section IV: The Propositions

What is a “proposition”? Using only what we read contextually from Euclid, a proposition seems to be something constructed through what is already there. But while they rely on something already established, they also create something new. Once the proposition is satisfactorily constructed (whether it is “quod erat demonstrandum” or “quod erat faciendum”) it is now part of our building blocks and can be used to construct something more.

As to the specific propositions, the overarching theme is simple to complex. More building blocks are available as we move along and Euclid makes use of them. Circles are the easiest to understand at first, so that is what Euclid uses first. But we must get to triangles for all the equalities we can learn from them. While circles are understandable, they are also much less useful than triangles since the only things we can relate them to are lines, points and triangles. Euclid never uses them in Book I for rectilineal angles, parallel lines, parallelograms, or squares. Yet triangles can be used to demonstrate relationships for all these shapes.

The trickiest part of Euclid’s first book seems to be the idea of parallel lines. While he gives them a meaning in Definition 23, he has to ask his reader to grant him some further concepts to support their existence in Postulate 5:

That if a straight line falling on two straight lines make the interior angles on that same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angle less than two right angles.

One could call this a clunky description. It is certainly inelegant when one compares it to the definition of a point or a line.

Euclid is also hesitant to use Postulate 5, waiting until Proposition 29 to reference it. At this point, he already has two propositions (27 and 28) that make use of parallel lines without using Postulate 5. So is it because he feels uncomfortable with the idea of parallel lines that he wants to get as many propositions constructed as possible before having to use them?

Here we find ourselves in a similar situation as when we finished the first two definitions. We are presented something which we must accept to continue on, but which we cannot really understand except in a hazy abstraction. So questions remain and relate to those from a point and a line: a) is Euclid creating an entire system that depends on us believing that things exist that we cannot prove exist and/or cannot physically comprehend, and b) do we accept a system that must be tied together using things that can’t exist in the physical universe as the starting point?

The answer to the former must be yes. He is asking us to either believe or at least agree with him on the definition of a point, a line, and that parallel lines will not meet. As to the latter, the most likely reply is “what is defined in abstraction must be agreed upon in abstraction.” Meaning that we must omit Euclid from the physical universe if he does cannot relate it to the physical universe.

Section V: Conclusion: Where does his structure leave us?

The only limit placed upon us implicitly at the end of Book I is “use what has come before.” Euclid has demonstrated that one can create a great deal out of a point, a line, and the relationship between the two. We can construct whatever we want from here, so long as it is not contradictory to what has come before.

But even if it is contradictory, we can examine our constructions and Euclid’s constructions exactly and determine which is more true. No conclusion is left unexplained and every step is laid out. So one can work within the bounds that Euclid has laid out and accept his definitions, postulates, common notions, and propositions and build on them. Or one can attempt to prove these wrong.

Porgi Amor

Uncategorized | Posted by Brian PCF
Oct 31 2013

The Countess’s Aria presents us for the first time in the opera, a seemingly uncomplicated character. Everyone so far has secrets and schemes. All of the characters want someone who wants someone else or wants something that someone else has, but the Countess just wants her husband, the man that should be hers.

The first things we notice about the Countess’s aria are the long intro. She has a full 17 measures of instrumentation that lays the mood for the piece before she begins her solo. What we here in that opening is strings: violin and viola, followed by clarinet, french horn, and bassoon. In the first few measures, the higher pitched violin and the lower pitched viola play together with no instrument taking the lead. The same goes for the first few measures when we have the higher pitched clarinet, with the lower pitched french horn and bassoon. But soon the lower pitched horns and viola sound softer and play less often, while the clarinet and violins continue to play louder and more often. The instrumentation seems to back up what we find later through the vocals: the Countess love lingers, but the Count’s no longer interested in what he’s already caught.

Next we notice the key: the same E-Flat we found in Cherubino’s aria in No. 6. His aria was “allegro vivace”, meaning “joyful, very lively tempo.” The Countess in “larghetto”, meaning “fairly slow tempo.” Also noteworthy is that Cherubino was on stage for much of Act I and never alone. The Countess is on for a mere 48 measures (the smallest part so far) and is singing by herself. Cherubino can’t really decide exactly who he wants in his aria, and his demand is that he either desperately wants love or he will “talk alone of love”.

The Countess words match the accompaniment: simple and to the point. The Countess wants what was seemingly promised (and perhaps at one time had), the love of her Count. And if she doesn’t get it, she would rather die. The instrumentation throughout her part is lacking in flourish. Many quarter notes in a very simple 2/4 time signature. Moreover, she just repeats the same refrain again and again.

This is seemingly the simplest and most straightforward character we’ve been introduced too, which makes for an interesting element. Every other character has been maneuvering and presenting one face to the audience, and another face to the other players. It is perhaps the most intriguing way to add complexity to the plot: add a simple, genuine person. The other characters won’t know how to handle her.

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Complete video here.
Complete score here.

On Euclid’s Definitions and Postulates

Uncategorized | Posted by Brian PCF
Oct 20 2013

The structure of Euclid’s definitions and postulates assumes a movement from simple to complex. From a beginning of the point, “that which has no part” we finish the postulates with a vast (possibly infinite in our modern understanding of the word) ability to create and explore points, lines, plane surfaces, and perhaps an approximation of our entire physical universe.

The definitions seem to build on each other in a seemingly logical form. We start with the point and line, and then expand on what a point and a line can bring about: straight lines, surfaces, plane angles, figures, circles, parallel lines. Literally, we can create from this ad infinitum. A basic proposition of Euclid seems to be that we can represent anything with this system of points and lines.

Interestingly though, we define these basic structures in the first two definitions in our Euclidean universe not by their properties, but by that which they are not: a point is “that which has no part”, a line “breadthless length”. We create representations of our physical universe from these definitions, but they can’t exist in our physical universe. Something that has no part, or no breadth or no depth can’t exist in reality as we understand it based on our senses.

Moreover, everything that comes after these basic definitions is relative to one another. There is no existence beyond the point and line in and of itself, simply (or perhaps more complicated, as we’ll see) is the idea that there are fundamental properties based on these relative relationships.

The first relative definition is the straight line, “a line which lies evenly with the points on itself.” Let’s assume that this means that “evenly” here means an angle of zero as we would define it. Euclid requires a much more flexible definition here as he doesn’t want to define a line with an angle zero. It seems from his Definition 8, “a plane angle is the inclination to one another of two lines in a plane which meet one another and do not line in a straight line” that he wants to have the ability to put as many points or lines on a single straight line as he needs. So a seemingly simple idea, but with an infinite amount of complexity that we can build in as needed.

We continue to see complex and relative figures emerge. Even something seemingly simple like a boundary, “that which is an extremity of anything”, or that of a figure, “that which is contained by any boundary or boundaries” depends upon some other thing for its existence and is hence relative.

Right angles are a fascinating example of this idea of relativity in that they are not only created by another right angle (definition 10), but are all exactly the same no matter what (postulate 4). So a right angle can’t come into being as a singular right angle, it must be born as a twin.

Circles are the next figure that require a relative relationship. They cannot exist in and of themselves, but require a center. The center also requires a circle, otherwise it is just a point. This and the right angle are all strikingly similar to “the swerve” of Lucretius: atoms colliding with one another to create the entire physical universe. While Euclid is contriving his universe, that idea of collision between basic atoms or ideas is still fundamental to the Elements.

Parallel lines are very similar to the idea of the right angle. They need each other to exist and they are fundamentally identical in that they are “breadthless length” and they share a relative angle.

So while nothing can exist in Euclid’s universe without points and lines, those can only exist by seemingly simple definitions that speak more to what these things are not versus what they are. Moreover, everything built from these points and lines is defined by their relative relationship with one another. There are no new things in Euclid, only these basic ideas of points and lines and how they interact. So the baseline is simple, but from this we can grow the whole of mathematics and approximate everything we see in the physical world.

[Using the Heath translation]

Camus on Vague Nostalgia and Unity

Uncategorized | Posted by Brian PCF
Jun 03 2013

Absurd Freedom

Now the main thing is done, I hold certain facts from which I cannot separate. What I know, what is certain, what I cannot deny, what I cannot reject – this is what counts. I can negate everything of that part of me that lives on vague nostalgias, except this desire for unity, this longing to solve, this need for clarity and cohesion. I can refute everything in the world surrounding me that offends or enraptures me, except this chaos, this sovereign chance and this divine equivalence which springs from anarchy. I don’t know whether this world has a meaning and that it is impossible for me just now to know it. What can a meaning outside my human condition mean to me? I can understand only in human terms. What I touch, what resists me – that is what I understand. And these two certainties – my appetite for the absolute and for the unity and the impossibility of reducing this world to a rational and reasonable principle – I also know that I cannot reconcile them. What other truth can I admit without lying, without brining in a hope I lack and which means nothing within the limits of my condition?

The Myth of Sisyphus, Camus

Nabokov on Freedom and Literature

Uncategorized | Posted by Brian PCF
Mar 21 2013

“It is difficult to refrain from the relief of irony, from the luxury of contempt, when surveying the mess that meek hands, obedient tentacles guided by the bloated octopus of the state, have managed to make out of the that fiery, fanciful free thing – literature. Even more: I have learned to treasure my disgust, because I know that by feeling so strongly about it I am saving what I can for the spirit of Russian literature. Next to the right to create, the right to criticize is the richest gift that liberty of thought and speech can offer. Living as you do in freedom, in that spiritual open where you were born and bred, you may be apt to regard stories of prison life coming from remote lands as exaggerated accounts spread by panting fugitives. That a country exists where for almost a quarter of a century literature has been limited to illustrating the advertisements of a firm of slave-traders is hardly credible to people for whom writing and reading books is synonymous with having and voicing individual opinions. But if you do not believe in the existence of such conditions, you may at least imagine them, and once you have imagined them you will realize with new purity and pride the value of real books written by free men for free men to read.” -Vladimir Nabokov, Lecture on Russian Literature.

 

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Nabokov on Guidebooks

Uncategorized | Posted by Brian PCF
Mar 20 2013

For an artist one consolation is that in a free country he is not actually forced to produce guidebooks. -Vladimir Nabokov

Found this gem in Vladimir Nabokov: Lectures on Russian Literature, which came on the heels of an interesting Writer’s Almanac discussing the New Deal Federal Writers Project, which employed writers during the depression and produced most notably guidebooks on the 48 states.

How To Start Your Own CrossFit Walter Reed

Uncategorized | Posted by Brian PCF
Mar 04 2013

The last several months it seems like more and more people are finding out about what we do at CrossFit Walter Reed and we are getting bombarded with emails asking us how we started the program and if we have any advice for starting a similar program for wounded veterans and/or active duty at other bases. The following is what we did and what I recommend you do.

Don’t ask anybody’s permission.

Dillon Behr, the founder of CrossFit Walter Reed, is a former Green Beret and patient at Walter Reed Army Hospital. He shadowed sessions with two injured vets that I was training at Potomac CrossFit, grabbed some guys that were still at the hospital, got his CrossFitting girlfriend who worked at the hospital to corral some more guys, and held a class.

He didn’t ask permission of the command, he didn’t ask permission of the gym. He used initiative and just did it.

This is the most important thing to remember in trying to start something like this up. Don’t ask anybody and don’t accept “no” as a response from anybody.*

“Be polite, be professional and have a plan to kill everyone you meet.” -Mattis

There will be people on base that love CrossFit and will sell their first born to help you out.

There will be others that think that CrossFit guarantees injury and is just another fitness fad that they need to keep their people from ever trying.

As you develop your program, you have a very delicate balancing act of trying to get enough attention that you can continue to attract athletes, but not so much attention that members of the command that might be antagonistic to your program find out about it.

What is critical is finding your “Rabbis”. These are folks with pull and/or rank within the command that when you have somebody that’s really inhibiting what you can do (and this usually comes in the form of crippling bureaucracy, not outright “no’s”), you can ask them to step in.

Charity

We are supported by a 501(c)3 called Team R4V which is a great organization that does everything they can for us.

We get the majority of our support from the U.S. Marine Corps Wounded Warrior Regiment (which is not affiliated with the Wounded Warrior Fund). Also, CrossFit HQ donated $25,000 worth of gear for our use.

If/when you start a CrossFit gym or program for Wounded Warriors, you are going to get your email blown up with offers to partner and/or help out from individuals and charities.

I certainly trust Team R4V, but with the vast number of people that have set up veteran’s charities and then pocketed most of the money, you have to vet who you work with. A great resource for this are GuideStar.Org. Every 501(c)3 has to post their tax returns on their website as well. So if you can’t find it, ask for a link to it and what you are trying to find out is what their fundraising amount was last year, what was the salary of their key personnel, and how much was spent on programs versus overhead.

If you can pull it off, get the command to roger up support and you don’t have to worry about this.

Coaches

Vetting should be foremost in your mind when selecting coaches too. You need to find people that are motivated, competent, and will show up week in and week out without flaking.

There are probably a million CrossFitters that love the “idea” of being a CrossFit Walter Reed coach, but experience has proved that only about six people can pull it off.

Most folks that have volunteered have un-volunteered. This doesn’t make them bad people. More or less we have this program covered, and an hour commuting and another hour of training in the middle of the day every week just isn’t workable for most folks.

When folks get in touch to help out, we have them just come in and workout with the guys. This gives us a chance to do some butt sniffing and gauge their competence and character. If they seem flaky or don’t seem qualified to coach, we tell them to keep coming in just to workout. This gauges their dedication and we can continue to screen.

Most folks figure out that they don’t have the time to come in and that the “Stand by Me” moments they were expecting aren’t the same as reality. So these folks just stop showing up.

The folks that demonstrate they are committed and competent get an assistant coach slot in one of our classes and we develop them from there.

Training

Oh yeah, you have to do that too.

This has been discussed plenty of time on our blog and on my resources post. What I’ll say briefly is this:

1) The plan only survives first enemy contact. You need to go with the flow because your workout with running and rowing will have 10 athletes and nine of them will be bilateral amputees that can’t run or row. Come up with a different plan and execute.

2) Don’t hurt these guys. Their bodies are spending a significant amount of energy healing itself. You need to give them a solid dose, or they won’t come back. But if you break them, they won’t come back either.

Hope this helps some folks, if you have questions, post to comments. And as always, essem plena cacas.

*If we ever have a drink, ask me about Marine Corps Community Services, LtCol Dan Wilson (who is an outstanding Marine Officer), and Operation Phoenix to find out how bad things can get with dealing with entrenched quasi-private interests.

Adaptive CrossFit Community and Resources

Uncategorized | Posted by Brian PCF
Mar 01 2013

We’re getting a ton of questions about what we call “Adaptive CrossFit” and I’m sending the same info to folks. Below is a consolidated list of POCs and resources to help folks out that have questions about this:

Dave “Chef” Wallach, CrossFit Rubicon, chef@cfrubicon.com.

Chef is, so far as I know, the most qualified Adaptive CrossFit coach on earth. He trains a bunch of amputees and TBI’s out of CrossFit Rubicon, has an amputee and TBI on his coaching staff, and has a very sophisticated technical understanding of exercise and injury. He was the organizer of the first ever Adaptive CrossFit competition last November at CrossFit Rubicon, the Working Wounded Games.

Brian Wilson (your humble author), Patriot CrossFit/Potomac CrossFit/CrossFit Walter Reed. brian@crossfitwalterreed.com.

I’m the hack that you can find taking up most of the screen time on Adaptive CrossFit videos. Outside of that I’m the Director of Operations and Co-Founder of CrossFit Walter Reed. In this roll, I take a bunch of credit while the CrossFit Walter Reed coaches Jason Sturm, Shad Lorenz, Danna Rey, Andrea Bates, Steve Michael, and volunteers do all the work (no, seriously).

If none of the above dissuaded you, you can still email me (or better yet post to comments!).

You can find several additional POCs on the Adaptive CrossFit thread on the CrossFit Message Board including Kim Hanna, Rick Martinez, and Kendra Bailey.

Here’s some resources to dig through.

Evolution of Adaptation
Adaptive Assets

A CrossFit Love Story
Adapt and Overcome: The Nick Thom Story

CFJ – Working Wounded by Greg Glassman
CFJ – Serving Soldiers by Brian Wilson
CrossFit Walter Reed website
Push, Pull, Lift, Press by Jon Gilson
Kyle Maynard at the Level I Cert (Day 1) (Day 2)
The Warrior Spirit I, II, III, IV
“Push Yourself Beyond Your Limits” – Hector Delgado

A Lucky Olympian

If there are other resources and POCs out there, please post to comments and I’ll put them in this post. If they aren’t in the above list, it’s not because I don’t like you or I think they’re dumb, I just don’t know they exist.